Answer:
It is valid.
Venn Diagram is attached.
I also provided a truth table.
I also provided a couple of names this argument could be called.
Please let me know if you have any questions about this question.
Thank you kindly.
Step-by-step explanation:
If a person is a poet, then that person is a loner.
If a person is a loner, then that person is a taxi driver.
If a person is a poet, then that person is a taxi drive.
This is in the form:
p->q
q->r
Therefore, p->r.
This is an example of a hypothetical syllogism.
The first thing my mind goes to when seeing this is the transitive property which says if A is related to B and B is related to C then A is related to C.
So this syllogism also has another name and it is called the principle of transitivity of implication. 
Let p=person is a poet, q=person is a loner, and r=person is a taxi driver.
I'm going to make a truth table and compare the truth values of the propositions to the conclusion.
Since we have 3 variables, we are going to need 2^3=8 rows.
p     q       r        p->q    q->r      (p->q) and (q->r)        conclusion=(p->r)
T      T     T           T         T                     T                               T
T      T     F           T         F                    F                                 F
T      F     T           F         T                     F                               T   
T      F     F           F         T                     F                                F
F      F     F           T         T                    T                                 T
F      F     T           T         T                    T                                 T
F     T     F            T         F                    F                                 T
F    T     T             T         T                    T                                 T
So we are trying to see if we ever have the preposition part true and the conclusion false.
That doesn't happen so the argument is valid.